Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.1591

Every sum system is divisible

Masaki Izumi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that every sum system is divisible. Combined with B. V. R. Bhat and R. Srinivasan's result, this shows that every product system arising from a sum system (and every generalized CCR flow) is either of type I or type III. A necessarily and sufficient condition for such a product system to be of type I is obtained.

Explore related subjects

math.OAmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Masaki Izumi. 2007-08-12. Every sum system is divisible. https://arxiv.org/abs/0708.1591

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A $\mathrm{C}^*$-algebraic Hoffman-Wielandt theorem

We observe that the $2$-norm distance $d_{U,2}$ between the unitary orbits of normal elements in a $\mathrm{II}_1$ factor $\mathcal{M}$ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace $τ_\mathcal{M}$. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\mathcal{Z}$-stable $\mathrm{C}^*$-algebras that are either monotracial, or real rank zero with finitely many extremal traces, provided that $σ(x)=σ(y)$ is convex. Consequently, $d_{U,2}$ equips the set of approximate unitary equivalence classes of contractive normal elements of $\mathcal{M}$ with the structure of a compact length space. The same is true of the set of equivalence classes of embeddings into the Jiang-Su algebra $\mathcal{Z}$ of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains.

math.OA↗
arXiv2026

The Cuntz semigroup of the graph C*-algebra of a finite graph

We compute the Cuntz semigroup of a graph C*-algebra associated to a finite graph by combining Hong and Szymański's realisation of the entire ideal lattice of a graph C*-algebra, Ara, Moreno and Pardo's computation of the Murray-von Neumann semigroup of a unital graph C*-algebra and various Cuntz semigroup techniques. We also investigate certain properties of such Cuntz semigroups, and demonstrate that at the level of the Cuntz semigroup, there is no obstruction to the conjecture that the nuclear dimension of any graph C*-algebra is at most one.

math.OA↗
arXiv2026

Quantum real projective spaces as quantum CW-complexes via topological graphs

We use the notion and properties of topological graphs to construct a quantum CW complex for quantum real projective spaces. This construction mirrors the CW complex construction in the classical case. Moreover, we introduce multipullback even-spheres as boundaries of noncommutative "cylinders" and show that they admit a gluing akin to the gluing of balls to the ends of a hollow closed cylinder.

math.OA↗